Möller, Joakim

Abstract [en]

In this thesis, we have investigated the Recursive Projection Method, RPM, as an accelerator for computations of both steady and unsteady flows, and as a stabilizer in a bifurcation analysis.

The criterion of basis extraction is discussed. It can be interpreted as a tolerance for the accuracy of the eigenspace spanned by the identified basis, alternatively it can be viewed as a criterion when the approximative Krylov sequence becomes numerically rank deficient.

Steady state calculations were performed on two different turbulent test-cases; a 2D supersonic nozzle flow with the Spalart-Allmaras 1-equation model and a 2D sub-sonic airfoil simulation using the κ - ε model. RPM accelerated the test-cases with a factor between 2 and 5.

In multi-scale problems, it is often of interest to model the macro-scale behavior, still retaining the essential features of the full systems. The ``coarse time stepper'' is a heuristic approach for circumventing the analytical derivation of models. The system studied here is a linear lattice of non-linear reaction sites coupled by diffusion. After reformulation of the time-evolution equation as a fixed-point scheme, RPM coupled with arc-length continuation is used to calculate the bifurcation diagrams of the effective (but analytically unavailable) equation.

Within the frame-work of dual time-stepping, a common approach in unsteady CFD-simulation, RPM is used to accelerate the convergence. Two test-cases were investigated; the von Karman vortex-street behind a cylinder at Re=100, and the periodic shock oscillation of a symmetric airfoil at M ∞ = 0.76 with a Reynolds number Re=11 x 106.

It was believed that once a basis had been identified, it could be retained for several steps. The simulations usually showed that the basis could only be retained for one step.

The need for updating the basis motivates the use of Krylov methods. The most common method is the (Block-) Arnoldi algorithm. As the iteration proceeds, Krylov methods become increasingly expensive and restart is required. Two different restart algorithm were tested. The first is that of Lehoucq and Maschhoff, which uses a shifted QR iteration, the second is a block extension of the single-vector Arnoldi method due to Stewart. A flexible hybrid algorithm is derived combining the best features of the two.

Runborg, Olof

Kevrekidis, P.G.

Lust, K.

Kevrekidis, I.G.

2005 (English)In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, ISSN 0218-1274, Vol. 15, no 3, 975-996 p.Article in journal (Refereed) Published

Abstract [en]

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Pade approximations.

Abstract [en]

The Recursive Projection Method (RPM) has been implemented into the unstructured grid CFD code EDGE to accelerate the inner-loop convergence of dual time stepping. The method tries to identify the slowly converging subspace and applies Newton iterations in this subspace together with a fixed point scheme in the complement. The method has been employed to compute the steady and unsteady viscous flow around a circular cylinder for a Reynolds number of 100. When converging to machine accuracy, RPM accelerated the convergence of the steady-state solution by a factor of 2.5. The time-accurate simulations were accelerated by a factor of about two.

Abstract [en]

The Recursive Projection Method (RPM) hasbeen implemented into an unstructured CFD code to improve the efficiency of dual time steppingfor unsteady turbulent CFD simulations.RPM is a combined implicit-explicit method that enhances convergence. It can easily be implementedinto existing codes and the solver’s existing acceleration techniques can be used withoutchange. The method has been evaluated by computing the periodic self-induced shock oscillations over an 18% thick biconvex airfoil at0◦ angle of attack, a Mach number of 0.76 anda Reynolds number of 11 million. On average,RPM accelerated the convergence of the innerloop of dual time stepping to a predefined convergencecriterion by a factor of about 2.5.